On Feb 9, 8:09 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:> On Feb 10, 10:19 am, Charlie-Boo <shymath...@gmail.com> wrote:>>>>>>>>>> > On Feb 7, 1:51 am, fom <fomJ...@nyms.net> wrote:>> > > On 2/5/2013 9:32 AM, Charlie-Boo wrote:>> > > > On Feb 4, 4:26 pm, fom <fomJ...@nyms.net> wrote:> > > >> On 2/4/2013 8:46 AM, Charlie-Boo wrote:>> > > >>> On Feb 4, 12:25 am, fom <fomJ...@nyms.net> wrote:> > > >>>> On 2/3/2013 10:19 PM, Charlie-Boo wrote:> > > >>>> <snip>>> > > >>>>>>>> In PROLOG we use lowercase words for TERMS> > > >>>>>>>> and uppercase words for VARIABLES>> > > >>>>>>>> ATOMIC PREDICATE>> > > >>>>>>> ATOMIC PREDICATE meaning relation?>> > > >>>>>>> C-B>> > > >>>>>> RELATION> > > >>>>>> p(a, b, e)>> > > >>>>> If wffs are built on relations then { x | x ~e x } is not a wff> > > >>>>> because ~e is not a relation.>> > > >>>> Well-formed formulas are built from the alphabet> > > >>>> of a formal language. If the language contains> > > >>>> a symbol of negation, then NOT(xex) will be a> > > >>>> well-formed formula.>> > > >>> You have to define what value a symbol may have - how it is> > > >>> interpreted in your definition of a wff. You need to complete B> > > >>> below to see there is no paradox if you are consistent about what a> > > >>> wff may contain and what values it may equal after substitution> > > >>> (interpretation) if it contains variables for functions.>> > > >> First, I was not in a good mood when I posted. So, I may> > > >> have been too dogmatic.>> > > >> What you seem to be objecting to is the historical development> > > >> of a logical calculus along the lines of Brentano and DeMorgan.>> > > I meant Bolzano here.>> > > > The only objecting in my Set Theory proposal is perhaps objecting to> > > > the fact that ZF has a dozen messy axioms, a dozen competing> > > > axiomatizations, a dozen interpretations of the most popular> > > > Axiomatization, and (Wikipedia), The precise meanings of the terms> > > > associated with the separation axioms has varied over time. The> > > > separation axioms have had a convoluted history, with many competing> > > > meanings for the same term, and many competing terms for the same> > > > concept.>> > > > (DeMorgan is an example of why Logic and Set Theory are the same thing> > > > and should be combined - same as Math and Computer Science etc.)>> > > How do you see Logic and Set Theory as being the same?>> > Both are concerned with mappings to {true,false}. A propositional> > calculus proposition is 0-place. A set is 1-place. A relation is any> > number of places. (A relation is a set - of tuples.)>> > So you have the same rules of inference: Double Negative, DeMorgan> > etc. apply to propositions and sets.>> > To prove incompleteness, Godel had to generalize wffs as expressing> > propositions to expressing sets when the wff has a free variable.>> > C-B>> Yes but you change the rules of the game depending what you want to> prove.>> Let:>> LANGUAGE 1 LANGUAGE 2>> seta(x) <-> x e seta>> proof(x) <-> x e proof>> proveby(x,y) <-> (x,y) e proveby>> russell(x) <-> not( x e x)>> ------------------->> In ZFC you "UNSTRATIFY" Russel's set> but in all theories > PA you necessitate Godel's Statment!>> Sheer blindness!>> TOM: Jerry can't say this sentence is true!> JERRY: Tom can't say this sentence is true!>> TOM AND JERRY ARE *INCOMPLETE* !!

There are a lot of premises there.

TOM can express "unprovable by JERRY".TOM can express truth in JERRY. (Uh-oh, Tarski)JERRY can express "unprovable by TOM".JERRY can express truth in TOM. (Uh-oh, Tarski)